March 21, 2022

Isoclines …

Separate phase space into areas of increasing and decreasing prey or predator abundance.

Lotka-Volterra

If isoclines are perpendicular … eternal oscillations.

Lotka-Volterra with Carrying Capacity

If isoclines tilts … damped oscillations

What if prey isocline rises and falls?

On the left … Allee effect and Prey switching

On the right … Density dependence

In the middle - stable oscillations

In the right - damped oscillations

In the left - unstable oscillations

So …. if you increase carrying capacity

You can send the system into an unstable state and INCREASE the probability of extinction.

Rosenzweig’s “enrichment paradox”

“…increasing the supply of limiting nutrients or energy tends to destroy the steady state. Thus man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in greater abundance.” - Rosenzweig (1971)

Nice theory you’ve got there ….

But is it real?

More experiments with Didinium nausutum vs. Paramecium caudatum.

High carrying capacity

Luckinbill 1973 recreates Gause experiments with a different food source.

Low carrying capacity

When the food source is limited … some nice oscillations!

Consistent with Enrichment Paradox …

Harrison (1995) reinterprets the data

x`

HOWEVER, in natural systems

efforts to show this effect generally fail. (reviewed in Roy and Chattopadhyay 2007).

Because things are complicated! Fluctuations are buffered by:

  • Anti-predation strategies
  • Multple prey species
  • Ratio-dependent responses
  • Refuge effect
  • Predation capacity

Stabilizing effect of refuge, graphically

As long as there is some predator free areas, the oscillations are stabilized.

Refuge effect

Example: Wildlife management “stumbled” in to this with wildlife reserves, the National Wildlife Refuge System, etc. On the one hand, people actively manage their land to attract game / fowl enrichment, on the other hand there are pockets where populations can always recover.

Stabilizing effect of predator carrying capacity

Embracing complexity

Lots of fundamental, heavily cited papers on the relative roles of predation, competition and external environmental effects on intertidal rocky communities.

Later (2008-2016) secretary of National Ocean and Atmosphere Administration (NOAA).

Introducing Functional Response

A functional response is the intake rate of a consumer as a function of food density.

A more realistic functional response

\[f(V) = {aV \over 1+ahV}\]

Where \(a\) is attack rate, \(h\) is handling time.

This assumes a single predator species focusing on a single prey species, getting slowed down by processing time.

Holling’s “disk” equation

An assistant picked sandpaper discs off a tackboard (Holling 1951).

Fundamental foundations for predation theory ensued.

Another functional response

Here, we assume that when prey are very scarce, there are other options for predators, or non-linear reasons why they’re harder to find.

Number of predators to prey abundance

Prey Response to Predation

Total Numerical Prey Response = Prey eaten / predator X predator response

Total Proportional Prey Response = Total prey response / Prey abundance

Prey deaths

This curve will (typically) peaks at some intermediate number, where:

  • (1) - N. Predators is (already) high,
  • (2) - N. of Prey consumed per Predator is high (approaching asymptote),

but

  • (3) - Total N of prey is NOT too high.

Prey growth

All of that is total mortality of prey. Now - what we need is proporional growth of prey. We have a good, time-tested model for that:

Combining Growth with Removal

Where the two lines intersect, Growth = Removal , we’re at equilibrium. The question is … are these equilibria stable?

Stability of equilibria

Follow the arrows!

As usual, super elegant theory … but what is it good for?

What limits moose?

Big Q: Are herbivore populations controlled bottom-up (food limited) or top-down (predator limited).

Models from theory

Food regulates

Models from theory

Food regulates, wolves limit

Models from theory

2 equilibria! Either food or wolves regulate.

Models from theory

Wolves definitely regulate.

Meta-analysis

Data aggregated from many sites with high and low moose and wolf densities.

Empirically estimated …

Functional response curve

Looks Type II-ish, which makes sense since moose is (often) the main prey of wolves, and wolves are (often) the main predator of moose.

Density dependent growth curve

Note: these data are only from Isle Royale (Lake Superior) long term study.

Put it all together …

  • low productive environment (3) system more likely to be predator limited

  • more productive environment (1) more likely to be food-limited

  • a 2-state model is unlikely ….

  • … but the numeric response curve is shallow. Meaning a local system (when perturbed) can switch from one to the other.

  • Consistent with observed variability not just in space but in time.